Find a unit vector perpendicular to each one of the vectors `vec(a)= ( 4 hat(i) - hat(j)+ 3 hat(k)) and vec(b)=(2hat(i) + 2 hat(j)- hat(k)).`

Explanation

We know that `(vec(a) xx vec(b))` is a vector perpendicular to each one of `vec (a) and vec(b)`. So, the required vector is `((vec(a) xx vec(b)))/(|vec(a) xx vec(b)|)`.
Now,`(vec(a) xx vec(b))= |(hat(i),hat(j), hat(k)), (4,-1,3),(2,2,-1)|`
`= (1-6) hat(i) - (-4-6) hat(j) + (8+ 2) hat(k)`
`=(-5hat(i) + 10 hat(j) + 10 hat(k))`.
`|vec(a) xxvec(b)|=sqrt((-5)^(2)+(10)^(2)+(10)^(2))=sqrt(225)=15.`
Hence, the required unit vector `=((-5hat(i)+10 hat(j)+10hat(k)))/15`
`=1/3(-hat(i)+2hat(j)+2 hat(k)).`